CS6840 - Modern Complexity Theory

Meetings  

• Theme 1 : Counting Complexity - 14 meetings

  • Meeting 01 : Mon, Jan 19, 08:00 am-08:50 am

  Administrative Announcements. Overview of the course. Counting Complexity. Computational Model. The class FP. Clique vs #Clique Problem. Cycle vs #Cycle Problem. Decision vs Counting Problems. Counting the number of spanning trees in a given undirected graph. Kirchoff's Matrix Tree Theorem. The need of a theory.

  References
  Exercises
  Reading

  Administrative Announcements. Overview of the course. Counting Complexity. Computational Model. The class FP. Clique vs #Clique Problem. Cycle vs #Cycle Problem. Decision vs Counting Problems. Counting the number of spanning trees in a given undirected graph. Kirchoff's Matrix Tree Theorem. The need of a theory.
  

  References

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  None

  • Meeting 02 : Tue, Jan 20, 12:00 pm-12:50 pm

  If #CYCLE is in FP, then P = NP. Discussion on #BPM problem. The Class #P. Closure properties.

  References
  Exercises
  Reading

  If #CYCLE is in FP, then P = NP. Discussion on #BPM problem. The Class #P. Closure properties.
  

  References

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  None

  • Meeting 03 : Tue, Jan 27, 12:00 pm-12:50 pm

  The bits and value of the #P function. Counting as a generalisation of NP, BPP, RP, PP and Parity P. NP is contained in PP. #P = FP if and only if PP = P.

  References
  Exercises
  Reading

  The bits and value of the #P function. Counting as a generalisation of NP, BPP, RP, PP and Parity P. NP is contained in PP. #P = FP if and only if PP = P.
  

  References

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  None

  • Meeting 04 : Tue, Jan 27, 06:00 pm-06:50 pm

  (Lecture on Jan 22nd, rescheduled to Jan 27, 6pm.)<br> Notion of #P-completeness. Attempt 1 : Parsimonious reductions. #SAT is #P-complete. #IND-SET is #P-complete. Shortcomings of the theory. General oracle query model of #P-completeness. Valiant's theorem.

  References
  Exercises
  Reading

  (Lecture on Jan 22nd, rescheduled to Jan 27, 6pm.)
  Notion of #P-completeness. Attempt 1 : Parsimonious reductions. #SAT is #P-complete. #IND-SET is #P-complete. Shortcomings of the theory. General oracle query model of #P-completeness. Valiant's theorem.
  

  References

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  None

  • Meeting 05 : Wed, Jan 28, 06:00 pm-06:50 pm

  (Lecture on Jan 22, rescheduled to Jan 28, 6 pm)<br> Determinant and permanent. Combinatorial interpretation of 0-1 permanent in terms of perfect matchings. Combinatorial interpretations of permanent over integers via cycle covers in graphs with integer weights on edges. A useful trick of "cancellations" using negative weights.

  References
  Exercises
  Reading

  (Lecture on Jan 22, rescheduled to Jan 28, 6 pm)
  Determinant and permanent. Combinatorial interpretation of 0-1 permanent in terms of perfect matchings. Combinatorial interpretations of permanent over integers via cycle covers in graphs with integer weights on edges. A useful trick of "cancellations" using negative weights.
  

  References

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  None

  • Meeting 06 : Thu, Jan 29, 11:00 am-11:50 am

  Permanent over Integers is #P-complete. Valiant's reduction.

  References
  Exercises
  Reading

  Permanent over Integers is #P-complete. Valiant's reduction.
  

  References

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  None

  • Meeting 07 : Fri, Jan 30, 10:00 am-10:50 am

  Permanent over {0,1} is #P-complete.

  References
  Exercises
  Reading

  Permanent over {0,1} is #P-complete.
  

  References

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  None

  • Meeting 08 : Mon, Feb 02, 08:00 am-08:50 am

  Viewing randomised Turing machines and complexity classes based on counting. Recal of BPP, RP, PP classes. Amplication lemma, for one-sided and two-sided error versions.

  References
  Exercises
  Reading

  Viewing randomised Turing machines and complexity classes based on counting. Recal of BPP, RP, PP classes. Amplication lemma, for one-sided and two-sided error versions.
  

  References

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  None

  • Meeting 09 : Tue, Feb 03, 12:00 pm-12:50 pm

  Valiant-Vazirani Lemma and its implications. Amplification. USAT vs Parity-SAT. Recap of Hash families and pairwise independence. Construction of a pairwise independent hash families.

  References
  Exercises
  Reading

  Valiant-Vazirani Lemma and its implications. Amplification. USAT vs Parity-SAT. Recap of Hash families and pairwise independence. Construction of a pairwise independent hash families.
  

  References

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  None

  • Meeting 10 : Thu, Feb 05, 11:00 am-11:50 am

  SAT reduces via a 1-sided randomised reduction to Parity-P with 1/n probability of success. SAT reduces via 2-sided-bounded-error randomized reduction to Parity-P. Viewing Parity P as a modulus constraint. Modulus amplification. Plan for the rest of the proof.

  References
  Exercises
  Reading

  SAT reduces via a 1-sided randomised reduction to Parity-P with 1/n probability of success. SAT reduces via 2-sided-bounded-error randomized reduction to Parity-P. Viewing Parity P as a modulus constraint. Modulus amplification. Plan for the rest of the proof.
  

  References

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  None

  • Meeting 11 : Thu, Feb 05, 06:30 pm-07:30 pm

  Generalising the argument to show PH reduces 2-sided-error randomised reductions to Parity-P. Viewing Parity P as a modulus constraint. Modulus amplication. Plan for the rest of the proof.

  References
  Exercises
  Reading

  Generalising the argument to show PH reduces 2-sided-error randomised reductions to Parity-P. Viewing Parity P as a modulus constraint. Modulus amplication. Plan for the rest of the proof.
  

  References

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  None

  • Meeting 12 : Mon, Feb 09, 08:00 am-08:50 am

  Modulus amplification. Toda's polynomials. Final P^#P Algorithm to complete Toda's theorem.

  References
  Exercises
  Reading

  Modulus amplification. Toda's polynomials. Final P^#P Algorithm to complete Toda's theorem.
  

  References

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  None

  • Meeting 13 : Tue, Feb 10, 12:00 pm-12:50 pm

  BPP is contained in Sigma_2.

  References
  Exercises
  Reading

  BPP is contained in Sigma_2.
  

  References

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  None

  • Meeting 14 : Thu, Feb 12, 11:00 am-11:50 am

  Second consequence of Amplification. One random string works for all. Advice classes. P/poly. It contains all unary (even undecidable !) languages.

  References
  Exercises
  Reading

  Second consequence of Amplification. One random string works for all. Advice classes. P/poly. It contains all unary (even undecidable !) languages.
  

  References

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  None

• Theme 2 : Circuit Complexity - 34 meetings

  • Meeting 15 : Tue, Feb 17, 12:00 pm-12:50 pm

  Languages and Boolean function families. Circuit model of computation. Circuit families. Language acceptance, and parameters of circuit such as size and depth.

  References
  Exercises
  Reading

  Languages and Boolean function families. Circuit model of computation. Circuit families. Language acceptance, and parameters of circuit such as size and depth.
  

  References

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  None

  • Meeting 16 : Wed, Feb 18, 06:00 pm-07:00 pm

  (Compensating for Feb 18th) <br> PSIZE=P/poly. Uniformity in circuit complexity.

  References	Class Notes
  Exercises
  Reading

  (Compensating for Feb 18th)
  PSIZE=P/poly. Uniformity in circuit complexity.
  

  References

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  Class Notes

  • Meeting 17 : Thu, Feb 19, 11:00 am-11:50 am

  Karp-Lipton Collapse Theorem<br> Meyer's Collapse Theorem

  References	Notes by <a href="https://lucatrevisan.wordpress.com/2010/04/28/cs245-lecture-6-karp-lipton/">Luca Trevisan</a>.
  Exercises
  Reading

  Karp-Lipton Collapse Theorem
  Meyer's Collapse Theorem
  

  References

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  Notes by Luca Trevisan.

  • Meeting 18 : Fri, Feb 20, 10:00 am-10:50 am

  Circuit lower bound approach to P vs NP. Are there any hard functions that requires large size? Over what basis? Post's characterization of a complete basis.

  References
  Exercises
  Reading

  Circuit lower bound approach to P vs NP. Are there any hard functions that requires large size? Over what basis? Post's characterization of a complete basis.
  

  References

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  None

  • Meeting 19 : Mon, Feb 23, 08:00 am-08:50 am

  Shannon's Counting Argument to show the existence of hard functions. Lupanov; matching upper bound. Formulas vs circuits.

  References
  Exercises
  Reading

  Shannon's Counting Argument to show the existence of hard functions. Lupanov; matching upper bound. Formulas vs circuits.
  

  References

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  None

  • Meeting 20 : Tue, Feb 24, 12:00 pm-12:50 pm

  Explicit function examples. PARITY, ADD(n,2), MAJ. Carry look-ahead adder as a constant-depth polynomial-size circuit. Class AC^0 and NC^1. NC-hierarchy and AC-hierarchy.

  References
  Exercises
  Reading

  Explicit function examples. PARITY, ADD(n,2), MAJ. Carry look-ahead adder as a constant-depth polynomial-size circuit. Class AC^0 and NC^1. NC-hierarchy and AC-hierarchy.
  

  References

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  None

  • Meeting 21 : Thu, Feb 26, 11:00 am-11:50 am

  NC hierarchy. Undecidability and NC^0. Logspace uniformity. L-uniform PSIZE = P. Evaluating NC^1 circuits in Logspace. L-uniform NC^1 is contained in L.

  References
  Exercises
  Reading

  NC hierarchy. Undecidability and NC^0. Logspace uniformity. L-uniform PSIZE = P. Evaluating NC^1 circuits in Logspace. L-uniform NC^1 is contained in L.
  

  References

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  None

  • Meeting 22 : Fri, Feb 27, 10:00 am-10:50 am

  Circuit version of Savitch's algorithm. NL is contained in L-uniform AC^1. Classes SAC^1, Monotone circuits in AC^1.

  References	Class notes
  Exercises
  Reading

  Circuit version of Savitch's algorithm. NL is contained in L-uniform AC^1. Classes SAC^1, Monotone circuits in AC^1.
  

  References

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  Class notes

  • Meeting 23 : Mon, Mar 02, 08:00 am-08:50 am

  Spira's depth reduction. Balancing Boolean formulas. Polynomial size Boolean formulas characterize NC^1. Monotone version of the depth reduction.

  References	Section 6.1, Page 159 of Jukna's Book.
  Exercises
  Reading

  Spira's depth reduction. Balancing Boolean formulas. Polynomial size Boolean formulas characterize NC^1. Monotone version of the depth reduction.
  

  References

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  Section 6.1, Page 159 of Jukna's Book.

  • Meeting 24 : Tue, Mar 03, 12:00 pm-12:50 pm

  Polysize Skew-circuits characterise NL.

  References
  Exercises
  Reading

  Polysize Skew-circuits characterise NL.
  

  References

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  None

  • Meeting 25 : Thu, Mar 05, 11:00 am-11:50 am

  AND, PARITY, MAJ, Threshold Function. Placing ADD(n,n) in NC^1. Offman's trick. Constant depth reductions. Reducing MULT(n,2) to ADD(n,n).

  References
  Exercises
  Reading

  AND, PARITY, MAJ, Threshold Function. Placing ADD(n,n) in NC^1. Offman's trick. Constant depth reductions. Reducing MULT(n,2) to ADD(n,n).
  

  References

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  None

  • Meeting 26 : Fri, Mar 06, 10:00 am-10:50 am

  Adding loglog n numbers of n bits each in AC^0. Extending it add log n number of n bits each in AC^0. Reductions from Threshold to MAJ, ADD(n,n) to BCOUNT, BCOUNT to MULT(n,2). Defining the TC^0 class. ACC^0 class. Their counterparts in the levels of the NC hierarchy.

  References
  Exercises
  Reading

  Adding loglog n numbers of n bits each in AC^0. Extending it add log n number of n bits each in AC^0. Reductions from Threshold to MAJ, ADD(n,n) to BCOUNT, BCOUNT to MULT(n,2). Defining the TC^0 class. ACC^0 class. Their counterparts in the levels of the NC hierarchy.
  

  References

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  None

  • Meeting 27 : Mon, Mar 09, 08:00 am-08:50 am

  BCOUNT reduces to Threshold. All symmetric functions are in NC^1 (in fact TC^0). Two statements without proof - Polylog threshold is in AC^0 and MULT(n,n) is in NC^1. Circuit lowerbound problem. Discussion on explicitness. Best known lower bounds. Roadmap.

  References
  Exercises
  Reading

  BCOUNT reduces to Threshold. All symmetric functions are in NC^1 (in fact TC^0). Two statements without proof - Polylog threshold is in AC^0 and MULT(n,n) is in NC^1. Circuit lowerbound problem. Discussion on explicitness. Best known lower bounds. Roadmap.
  

  References

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  None

  • Meeting 28 : Tue, Mar 10, 12:00 pm-12:50 pm

  Gate Elimination Arguments. 2n-4 Lower bound for Threshold and 3(n-3) Lower bound for PARITY.

  References	Section 1.6, Page 39 of Jukna's Book
  Exercises
  Reading

  Gate Elimination Arguments. 2n-4 Lower bound for Threshold and 3(n-3) Lower bound for PARITY.
  

  References

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  Section 1.6, Page 39 of Jukna's Book

  • Meeting 29 : Thu, Mar 12, 11:00 am-11:50 am

  Sabbotavskaya's restriction method. Superlinear lower bound for PARITY. Universal function (addressing function) and Nechiporuk's lower bound for the same.

  References	Section 6.3 (page 167) in Jukna's Book
  Exercises
  Reading

  Sabbotavskaya's restriction method. Superlinear lower bound for PARITY. Universal function (addressing function) and Nechiporuk's lower bound for the same.
  

  References

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  Section 6.3 (page 167) in Jukna's Book

  • Meeting 30 : Fri, Mar 13, 10:00 am-10:50 am

  Density version of Sabbotavskaya's argument. Expected reduction in size of the formula. Shinkage exponent. Andreev's lower bound (statement). Nechiporuk's lower bound for the universal function with OR as the combining function.

  References	Section 6.2 (page 164) in Jukna's Book
  Exercises
  Reading

  Density version of Sabbotavskaya's argument. Expected reduction in size of the formula. Shinkage exponent. Andreev's lower bound (statement). Nechiporuk's lower bound for the universal function with OR as the combining function.
  

  References

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  Section 6.2 (page 164) in Jukna's Book

  • Meeting 31 : Mon, Mar 16, 08:00 am-08:50 am

  Andreev's Cubic Lower Bound. Effect of shrinkage exponent. Statement of Nechiporuk's subfunction method.

  References	Section 6.3 (page 168) in Jukna's Book
  Exercises
  Reading

  Andreev's Cubic Lower Bound. Effect of shrinkage exponent. Statement of Nechiporuk's subfunction method.
  

  References

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  Section 6.3 (page 168) in Jukna's Book

  • Meeting 32 : Tue, Mar 17, 12:00 pm-12:50 pm

  Nechiporuk's subfunction argument. Lower bound for element distinctness.

  References	Section 6.5 (page 173) in Jukna's Book
  Exercises
  Reading

  Nechiporuk's subfunction argument. Lower bound for element distinctness.
  

  References

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  Section 6.5 (page 173) in Jukna's Book

  • Meeting 33 : Thu, Mar 19, 11:00 am-11:50 am

  Lower bounds strategy for PARITY against AC^0. Effect of random restriction. A notion of "simplification". Decision tree model. PARITY, AND, OR. Size and depth of DTs. DT as a measure of Boolean functions. Lower bounds and Upper Bounds. DNFs. Decision Trees to DNFs, Decision trees to Skew Formulas. Termsize of DNFs for parity. From DNFs to Decision trees. Canonical decision trees.

  References
  Exercises
  Reading

  Lower bounds strategy for PARITY against AC^0. Effect of random restriction. A notion of "simplification". Decision tree model. PARITY, AND, OR. Size and depth of DTs. DT as a measure of Boolean functions. Lower bounds and Upper Bounds. DNFs. Decision Trees to DNFs, Decision trees to Skew Formulas. Termsize of DNFs for parity. From DNFs to Decision trees. Canonical decision trees.
  

  References

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  • Meeting 34 : Fri, Mar 20, 10:00 am-10:50 am

  Recap of canonical decision trees from DNFs. Hastad's switching Lemma. Using switching lemma to prove that any depth-2 circuit for PARITY requires exponential size.

  References
  Exercises
  Reading

  Recap of canonical decision trees from DNFs. Hastad's switching Lemma. Using switching lemma to prove that any depth-2 circuit for PARITY requires exponential size.
  

  References

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  • Meeting 35 : Mon, Mar 23, 08:00 am-08:50 am

  Extending the restrictions to higher depths. Induction on the depth. Final lower bound for parity assuming the switching lemma.

  References
  Exercises
  Reading

  Extending the restrictions to higher depths. Induction on the depth. Final lower bound for parity assuming the switching lemma.
  

  References

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  • Meeting 36 : Tue, Mar 24, 12:00 pm-12:50 pm

  Razoborov's proof of Hastad's Switching Lemma. The injective map from bad restrictions to stars, diff and stricter restrictions. Combinatorial bounds for stars(r,s).

  References
  Exercises
  Reading

  Razoborov's proof of Hastad's Switching Lemma. The injective map from bad restrictions to stars, diff and stricter restrictions. Combinatorial bounds for stars(r,s).
  

  References

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  • Meeting 37 : Thu, Mar 26, 11:00 am-11:50 am

  Polynomial method. Representing Boolean functions using polynomials. Degree and Sparsity. MAJ not in AC^0[2] and PARITY not in AC^0[3]. Overview. Approximating OR.

  References
  Exercises
  Reading

  Polynomial method. Representing Boolean functions using polynomials. Degree and Sparsity. MAJ not in AC^0[2] and PARITY not in AC^0[3]. Overview. Approximating OR.
  

  References

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  • Meeting 38 : Mon, Mar 30, 08:00 am-08:50 am

  Approximating OR. From a circuit to a polynomial approximation.

  References
  Exercises
  Reading

  Approximating OR. From a circuit to a polynomial approximation.
  

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  • Meeting 39 : Mon, Mar 30, 10:00 am-10:50 am

  From a circuit to a polynomial approximation.

  References
  Exercises
  Reading

  From a circuit to a polynomial approximation.
  

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  • Meeting 40 : Tue, Mar 31, 12:00 pm-12:50 pm

  Degree bounds for approximating MAJORITY. MAJORITY is not in AC^0. Degree bounds for approximating PARITY. PARITY not AC^0[3]

  References
  Exercises
  Reading

  Degree bounds for approximating MAJORITY. MAJORITY is not in AC^0. Degree bounds for approximating PARITY. PARITY not AC^0[3]
  

  References

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  • Meeting 41 : Wed, Apr 01, 08:00 am-08:50 am

  Monotone circuit complexity. Razborov's lower bound. The O(log n) negations barrier. Approximators. Inductive construction of approximators. Positive inputs and negative inputs.

  References
  Exercises
  Reading

  Monotone circuit complexity. Razborov's lower bound. The O(log n) negations barrier. Approximators. Inductive construction of approximators. Positive inputs and negative inputs.
  

  References

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  • Meeting 42 : Thu, Apr 02, 11:00 am-11:50 am

  No class

  References
  Exercises
  Reading

  No class
  

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  • Meeting 43 : Fri, Apr 03, 10:00 am-10:50 am

  Holiday

  References
  Exercises
  Reading

  Holiday
  

  References

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  • Meeting 44 : Mon, Apr 06, 08:00 am-08:50 am

  How good is the approximator circuit? Proof that there must be lots of errors in the sample inputs.

  References
  Exercises
  Reading

  How good is the approximator circuit? Proof that there must be lots of errors in the sample inputs.
  

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  • Meeting 45 : Tue, Apr 07, 12:00 pm-12:50 pm

  Argument for an upper bound for the errors on the positive and the negative inputs. Proof of the Sunflower Lemma.

  References
  Exercises
  Reading

  Argument for an upper bound for the errors on the positive and the negative inputs. Proof of the Sunflower Lemma.
  

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  • Meeting 46 : Thu, Apr 09, 11:00 am-11:50 am

  Round up of circuit lower techniques - Gate elimination argument, restriction method, switchin lemma, polynomial method, clique approximator method. Coming up : Algorithmic method and derandomization method.<br><br> Offshoot from decision trees - Branching program model. Examples. Conjecture about Bounded Width Branching Programs

  References
  Exercises
  Reading

  Round up of circuit lower techniques - Gate elimination argument, restriction method, switchin lemma, polynomial method, clique approximator method. Coming up : Algorithmic method and derandomization method.
  
  Offshoot from decision trees - Branching program model. Examples. Conjecture about Bounded Width Branching Programs
  

  References

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  • Meeting 47 : Fri, Apr 10, 10:00 am-10:50 am

  Barrington's surprising theorem and the proof idea. From BPs to Progams over monoids. From programs over monoids to BPs. Constructing programs with specific 5-cycle as accepting elements.

  References
  Exercises
  Reading

  Barrington's surprising theorem and the proof idea. From BPs to Progams over monoids. From programs over monoids to BPs. Constructing programs with specific 5-cycle as accepting elements.
  

  References

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  • Meeting 48 : Mon, Apr 13, 08:00 am-08:50 am

  Proof of Barrington's theorem.

  References
  Exercises
  Reading

  Proof of Barrington's theorem.
  

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• Theme 3 : Hardness vs Randomness - 5 meetings

  • Meeting 49 : Tue, Apr 14, 12:00 pm-12:50 pm

  Hardness vs Randomness regime. Deranomization problem. Pseudorandom distributions. Pseudorandom Generators. Using PRGs to derandomize BPPs.

  References
  Exercises
  Reading

  Hardness vs Randomness regime. Deranomization problem. Pseudorandom distributions. Pseudorandom Generators. Using PRGs to derandomize BPPs.
  

  References

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  • Meeting 50 : Thu, Apr 16, 11:00 am-11:50 am

  Worst-case hardness and average-case hardness of functions. Hardness amplification. Overview of how codes will help in hardness amplification

  References
  Exercises
  Reading

  Worst-case hardness and average-case hardness of functions. Hardness amplification. Overview of how codes will help in hardness amplification
  

  References

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  • Meeting 51 : Fri, Apr 17, 10:00 am-10:50 am

  Yao's unpredictability implies the pseudorandomness lemma. One-bit PRG, Two-bit PRG from polynomial average-case hardness bounds.

  References
  Exercises
  Reading

  Yao's unpredictability implies the pseudorandomness lemma. One-bit PRG, Two-bit PRG from polynomial average-case hardness bounds.
  

  References

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  • Meeting 52 : Mon, Apr 20, 08:00 am-08:50 am

  Nisan-Wigderson generator.

  References
  Exercises
  Reading

  Nisan-Wigderson generator.
  

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  None

  • Meeting 53 : Tue, Apr 21, 12:00 pm-12:50 pm

  References
  Exercises
  Reading

  
  

  References

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• Theme 4 : Interactive Proofs - 7 meetings

  • Meeting 54 : Thu, Apr 23, 11:00 am-11:50 am

  Interactive Proof Systems. Deterministic, Randomized. <br>Power of Interaction 1 : Interactive Proof System for GNI. Power of Interaction 2: Making BPP one sided error using interaction. Public Coin Protocols. Classes AM and MA. Structural theorems: MA is in Sigma_2. Completeness for MA can be improved to 1. MA is contained in AM. Completeness for AM can be improved to 1. AM is contained in Pi_2. If CoNP is in AM, PH collapses to AM. Set lower bound protocol.Public Coin Protocol for GNI. GNI is in AM. Set lower bound protocol. Hence if GI is NP-complete, PH = Sigma_2. Protocol for permanent. LFKN protocol and analysis. Protocol for #SAT. Generalization to prove IP = PSPACE.

  References
  Exercises
  Reading

  Interactive Proof Systems. Deterministic, Randomized.
  Power of Interaction 1 : Interactive Proof System for GNI. Power of Interaction 2: Making BPP one sided error using interaction. Public Coin Protocols. Classes AM and MA. Structural theorems: MA is in Sigma_2. Completeness for MA can be improved to 1. MA is contained in AM. Completeness for AM can be improved to 1. AM is contained in Pi_2. If CoNP is in AM, PH collapses to AM. Set lower bound protocol.Public Coin Protocol for GNI. GNI is in AM. Set lower bound protocol. Hence if GI is NP-complete, PH = Sigma_2. Protocol for permanent. LFKN protocol and analysis. Protocol for #SAT. Generalization to prove IP = PSPACE.
  

  References

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  • Meeting 55 : Fri, Apr 24, 10:00 am-10:50 am

  Arithmetization of QBF. Properties of Arithmetization. Mindblocks with respect to the protocol #SAT protocol. Degree Reduction using Quantifier introduction. Examples. Chinese remaindering. Coefficient size reduction. Completing Shamir's Proof of PSPACE is contained in IP.

  References
  Exercises
  Reading

  Arithmetization of QBF. Properties of Arithmetization. Mindblocks with respect to the protocol #SAT protocol. Degree Reduction using Quantifier introduction. Examples. Chinese remaindering. Coefficient size reduction. Completing Shamir's Proof of PSPACE is contained in IP.
  

  References

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  • Meeting 56 : Mon, Apr 27, 08:00 am-08:50 am

  IP is contained in PSPACE.

  References
  Exercises
  Reading

  IP is contained in PSPACE.
  

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  • Meeting 57 : Tue, Apr 28, 12:00 pm-12:50 pm

  The setting of Inapproximability. Independent Set Problem. GapIS. Connection between GapIS problem and inapproximability of IS. GapSAT and q-GapCSP. From PCP theorem to hardness of q-GapCSP.

  References
  Exercises
  Reading

  The setting of Inapproximability. Independent Set Problem. GapIS. Connection between GapIS problem and inapproximability of IS. GapSAT and q-GapCSP. From PCP theorem to hardness of q-GapCSP.
  

  References

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  • Meeting 58 : Thu, Apr 30, 11:00 am-11:50 am

  L is in PCP(O(log n),q) if and only if L reduces to q-GapCSP. Statement of PCP theorem. Hardness of GapSAT. Hardness of GapIS. Constant factor inapproximability for Independent Set problem.

  References
  Exercises
  Reading

  L is in PCP(O(log n),q) if and only if L reduces to q-GapCSP. Statement of PCP theorem. Hardness of GapSAT. Hardness of GapIS. Constant factor inapproximability for Independent Set problem.
  

  References

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  • Meeting 59 : Fri, May 01, 10:00 am-10:50 am

  Weak PCP for LIN. Random vector testing. Allowing exponential sized proofs. BLR Test for linearity. Correctness Proofs.

  References
  Exercises
  Reading

  Weak PCP for LIN. Random vector testing. Allowing exponential sized proofs. BLR Test for linearity. Correctness Proofs.
  

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  • Meeting 60 : Tue, May 05, 12:00 pm-12:50 pm

  Remarks about generalizing to setting of Qudratic Equations. Proof of consistency. Dinur's PCP Theorem. Statement and outline of the proof.

  References
  Exercises
  Reading

  Remarks about generalizing to setting of Qudratic Equations. Proof of consistency. Dinur's PCP Theorem. Statement and outline of the proof.
  

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